Comment on "Black holes in $f(\mathbb{Q})$ gravity"

In the work [Phys.Rev.D 105 (2022) 2, 024042], D'Ambrosio et al. investigated spherically symmetric black hole solutions in $f(\mathbb{Q})$ gravity, where several solutions satisfy the condition: $g_{tt}g_{rr} = \mathrm{const}$. This condition is characteristic of many black holes, including the Schwarzschild spacetime. In this Comment, we argue that no nontrivial vacuum black hole solutions satisfy this condition in $f(\mathbb{Q})$ gravity. We demonstrate our claim by reexamining the field equations under the "Set 2" connection called by D'Ambrosio et al., which is necessary for obtaining solutions distinct from those of general relativity (GR). For the case where the free parameters $c$ and $k$ are zero, i.e., Option 2 in their work, we show that any attempt to find a solution beyond GR forces the non-metricity scalar to vanish ($\mathbb{Q}=0$), which trivializes the field equations and does not describe a valid black hole solution. Our findings indicate that the condition, $g_{tt}g_{rr} = \mathrm{const.}$, is overly restrictive for finding new, static and spherically symmetric vacuum black holes in $f(\mathbb{Q})$ gravity. This conclusion does not depend on the specific form of $f(\mathbb{Q})$. We also briefly discuss Option 1 that was not addressed in D'Ambrosio et al.'s work, and give new constraints for the selection of parameters $c$ and $k$.